Abstract
In many mixture-process experiments, restricted randomization occurs and split-plot designs are commonly employed to handle these situations. The objective of this study was to obtain an optimal split-plot design for performing a mixture-process experiment. A split-plot design composed of a combination of a simplex centroid design of three mixture components and a 2<sup>2</sup> factorial design for the process factors was assumed. Two alternative arrangements of design points in a split-plot design were compared. Design-Expert® version 10 software was used to construct I-and D-optimal split-plot designs. This study employed A-, D-, and E- optimality criteria to compare the efficiency of the constructed designs and fraction of design space plots were used to evaluate the prediction properties of the two designs. The arrangement, where there were more subplots than whole-plots was found to be more efficient and to give more precise parameter estimates in terms of A-, D- and E-optimality criteria. The I-optimal split-plot design was preferred since it had the capacity for better prediction properties and precision in the measurement of the coefficients. We thus recommend the employment of split-plot designs in experiments involving mixture formulations to measure the interaction effects of both the mixture components and the processing conditions. In cases where precision of the results is more desirable on the mixtures as well as where the mixture blends are more than the sets of process conditions, we recommend that the mixture experiment be set up at each of the points of a factorial design. In situations where the interest is on prediction aspects of the system, we recommend the I-optimal split-plot design to be employed since it has low prediction variance in much of the design space and also gives reasonably precise parameter estimates.
Highlights
In many practical situations, the response for the mixture may not just depend on the mixture components and on experimental conditions that are referred to as the process variables
Split-plot design allows the researcher to model the effect of the mixture components and their process conditions simultaneously
We recommend the employment of split-plot designs in experiments involving mixture formulations to measure the interaction effects of both the mixture components and the processing conditions
Summary
The response for the mixture may not just depend on the mixture components and on experimental conditions that are referred to as the process variables. Mixture-process variable experiments are common in many fields such as food, chemical, pharmaceutical and processing industries. The process variables are not part of the mixture components but their levels when changed could affect the blending properties of the components [4]. The choice of the combination of the mixture design and the process variable design depend on the purpose of the design. Cornell and Vining [10] proposed a Split-Plot Design (SPD) for mixture-process variable experiments. The mixture-process variables in the SPD structure have two levels of randomization leading to two types of errors; the whole plot and the subplot errors. Montgomery [13] recommended splitplot design as the way to deal with restricted randomization
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